Statistical Methods for Discrete Response, Time Series, and Panel Data (W271): Lab 4

Instructions:

• \(\textbf{Due Date: 12/11/2018 (11:59 p.m. Pacific Time)}\)
• \(\textbf{Page limit of the pdf report: 20 (not include title and the table of content page}\)
• Use the margin, linespace, and font size specification below:
  • fontsize=11pt
  • margin=1in
  • line_spacing=single
• Submission:
  • Each group makes one submission to Github; please have one of your team members made the submission
  • Submit 2 files:
    1. A pdf file including the details of your analysis and all the R codes used to produce the analysis. Please do not suppress the codes in your pdf file.
    2. R markdown file used to produce the pdf file
  • Use the following file-naming convensation; fail to do so will receive 10% reduction in the grade:
    • FirstNameLastName1_FirstNameLastName2_FirstNameLastName3_LabNumber.fileExtension
    • For example, if you have three students in the group for Lab Z, and their names are Gerard Kelley, Steve Yang, and Jeffrey Yau, then you should name your file the following
      • GerardKelley_SteveYang_JeffreyYau_LabZ.Rmd
      • GerardKelley_SteveYang_JeffreyYau_LabZ.pdf
  • Although it sounds obvious, please write the name of each members of your group on page 1 of your pdf and Rmd files.

• This lab can be completed in a group of up to 3 students in your session. Students are encouraged to work in a group for the lab.

• For statistical methods that we cover in this course, use only the R libraries and functions that are covered in this course. If you use libraries and functions for statistical modeling that we have not covered, you have to provide (1) explanation of why such libraries and functions are used instead and (2) reference to the suppressWarnings(suppressMessages(library documentation. Lacking the explanation and reference to the documentation will result in a score of zero for the corresponding question.

• Students are expected to act with regards to UC Berkeley Academic Integrity.

---

Description of the Lab

In this lab, you are asked to answer the question “Do changes in traffic laws affect traffic fatalities?” To do so, you will conduct the tasks specified below using the data set driving.Rdata, which includes 25 years of data that cover changes in various state drunk driving, seat belt, and speed limit laws.

Specifically, this data set contains data for the 48 continental U.S. states from 1980 through 2004. Various driving laws are indicated in the data set, such as the alcohol level at which drivers are considered legally intoxicated. There are also indicators for “per se” laws—where licenses can be revoked without a trial—and seat belt laws. A few economics and demographic variables are also included. The description of the each of the variables in the dataset is come with the dataste.

Exercises:

1. Load the data. Provide a description of the basic structure of the dataset, as we have done throughout the semester. Conduct a very thorough EDA, which should include both graphical and tabular techniques, on the dataset, including both the dependent variable totfatrte and the potential explanatory variables. You need to write a detailed narrative of your observations of your EDA. Reminder: giving an “output dump” (i.e. providing a bunch of graphs and tables without description and hoping your audience will interpret them) will receive a zero in this exercise.

# Insert the function to *tidy up* the code when they are printed out
suppressWarnings(suppressMessages(library(knitr)))
opts_chunk$set(tidy.opts=list(width.cutoff=60),tidy=TRUE)
# Load libraries
suppressWarnings(suppressMessages(library(car)))
suppressWarnings(suppressMessages(library(dplyr)))
suppressWarnings(suppressMessages(library(Hmisc)))
suppressWarnings(suppressMessages(library(ggplot2)))
suppressWarnings(suppressMessages(library(lattice)))
suppressWarnings(suppressMessages(library(plm)))
suppressWarnings(suppressMessages(library(plyr)))

load("driving.RData")
#view data and descriptions
head(data, 5)

  year state sl55 sl65 sl70 sl75 slnone seatbelt minage zerotol gdl bac10 bac08 perse totfat
1 1980     1    1    0    0    0      0        0     18       0   0     1     0     0    940
2 1981     1    1    0    0    0      0        0     18       0   0     1     0     0    933
3 1982     1    1    0    0    0      0        0     18       0   0     1     0     0    839
4 1983     1    1    0    0    0      0        0     18       0   0     1     0     0    930
5 1984     1    1    0    0    0      0        0     18       0   0     1     0     0    932
  nghtfat wkndfat totfatpvm nghtfatpvm wkndfatpvm statepop totfatrte nghtfatrte wkndfatrte
1     422     236      3.20      1.437      0.803  3893888     24.14      10.84       6.06
2     434     248      3.35      1.558      0.890  3918520     24.07      11.08       6.33
3     376     224      2.81      1.259      0.750  3925218     21.37       9.58       5.71
4     397     223      3.00      1.281      0.719  3934109     23.64      10.09       5.67
5     421     237      2.83      1.278      0.720  3951834     23.58      10.65       6.00
  vehicmiles unem perc14_24 sl70plus sbprim sbsecon d80 d81 d82 d83 d84 d85 d86 d87 d88 d89 d90
1   29.37500  8.8      18.9        0      0       0   1   0   0   0   0   0   0   0   0   0   0
2   27.85200 10.7      18.7        0      0       0   0   1   0   0   0   0   0   0   0   0   0
3   29.85765 14.4      18.4        0      0       0   0   0   1   0   0   0   0   0   0   0   0
4   31.00000 13.7      18.0        0      0       0   0   0   0   1   0   0   0   0   0   0   0
5   32.93286 11.1      17.6        0      0       0   0   0   0   0   1   0   0   0   0   0   0
  d91 d92 d93 d94 d95 d96 d97 d98 d99 d00 d01 d02 d03 d04 vehicmilespc
1   0   0   0   0   0   0   0   0   0   0   0   0   0   0     7543.874
2   0   0   0   0   0   0   0   0   0   0   0   0   0   0     7107.785
3   0   0   0   0   0   0   0   0   0   0   0   0   0   0     7606.622
4   0   0   0   0   0   0   0   0   0   0   0   0   0   0     7879.802
5   0   0   0   0   0   0   0   0   0   0   0   0   0   0     8333.562

desc

       variable                                          label
1          year                              1980 through 2004
2         state            48 continental states, alphabetical
3          sl55                              speed limit == 55
4          sl65                              speed limit == 65
5          sl70                              speed limit == 70
6          sl75                              speed limit == 75
7        slnone                                 no speed limit
8      seatbelt     =0 if none, =1 if primary, =2 if secondary
9        minage                           minimum drinking age
10      zerotol                             zero tolerance law
11          gdl                  graduated drivers license law
12        bac10                        blood alcohol limit .10
13        bac08                        blood alcohol limit .08
14        perse administrative license revocation (per se law)
15       totfat                       total traffic fatalities
16      nghtfat                     total nighttime fatalities
17      wkndfat                       total weekend fatalities
18    totfatpvm         total fatalities per 100 million miles
19   nghtfatpvm     nighttime fatalities per 100 million miles
20   wkndfatpvm       weekend fatalities per 100 million miles
21     statepop                               state population
22    totfatrte        total fatalities per 100,000 population
23   nghtfatrte    nighttime fatalities per 100,000 population
24   wkndfatrte       weekend accidents per 100,000 population
25   vehicmiles               vehicle miles traveled, billions
26         unem                     unemployment rate, percent
27    perc14_24          percent population aged 14 through 24
28     sl70plus                           sl70 + sl75 + slnone
29       sbprim                     =1 if primary seatbelt law
30      sbsecon                   =1 if secondary seatbelt law
31          d80                             =1 if year == 1980
32          d81                                               
33          d82                                               
34          d83                                               
35          d84                                               
36          d85                                               
37          d86                                               
38          d87                                               
39          d88                                               
40          d89                                               
41          d90                                               
42          d91                                               
43          d92                                               
44          d93                                               
45          d94                                               
46          d95                                               
47          d96                                               
48          d97                                               
49          d98                                               
50          d99                                               
51          d00                                               
52          d01                                               
53          d02                                               
54          d03                                               
55          d04                             =1 if year == 2004
56 vehicmilespc                                               

This data set is composed of 1200 observations of 56 variables. Variable 22, totfatrte, is our outcome variable of interest. Variables 31-55 are dummy variables indicating the year.

# test observiations per state
min(table(data$state))

[1] 25

max(table(data$state))

[1] 25

Each state has exactly 25 obsearvations – panel is balanced.

par(mfrow=c(2,2))
hist(data$totfatrte)
hist(data$perc14_24)

hist(data$vehicmilespc)
hist(data$unem)

hist(data$bac08) 
hist(data$bac10) 

hist(data$perse) 
hist(data$sbprim) 

hist(data$sbsecon) 
hist(data$sl70plus) 

hist(data$gdl) 

Variables displays some degree of skewness and may need to transform. Additionally, some variables that appear to be binary have values between 0 and 1 to indicate mid year changes. We will round these variables.

ggplot(data, aes(x = year, y = totfatrte, group = as.factor(state))) +
  geom_boxplot() +
  ggtitle("Total fatalities per 100,000 population") + theme_bw()

States display a large difference in distribution of total fatality rate.

#ryan's plot
ggplot(data, aes(as.factor(year), totfatrte)) +
 geom_boxplot() +
 ggtitle('Total Fatalities by Year') +
 xlab('Year') + ylab('Total Fatalities') +
 theme(axis.text.x = element_text(angle = 45, hjust = 1.2))

# totfatrte: total fatalities per 100,000 population
ggplot(data, aes(x = year, y = totfatrte, colour = as.factor(state))) +
  geom_line(alpha = 0.7, show.legend = F) +
  ggtitle("Total fatalities per 100,000 population") + theme_bw()

Total fatalities decreased from 1980 to 1985 and remained roughly constant post 1985.

## viz variables to use
#every state throughout the years:
#totfartre, perc14_24, vehicmilespc and unem
#distribution across diff state: 
# seatbelt, zerotol, gdl, perse, bac08, bac01 and slXX variables
# if "1" in bac10, set as 0.1; else set as "0" or "1" in bac08
data$bac_combined <- ifelse(round(data$bac10) > 0, 0.1, 0.08*round(data$bac08))
data$sl_combined <- ifelse(round(data$sl55) > 0, 55, ifelse(round(data$sl65) > 0, 65,
                         ifelse(round(data$sl70) > 0, 70, ifelse(round(data$sl75) > 0, 75, 80))))

ggplot(data, aes(x = year, y = bac_combined, group = state)) + 
  geom_jitter(alpha = 0.3) + theme_bw() +
  ggtitle("Blood alcohol limit")

We noticed over time that fewer states have no BAC limit or 0.1 limit and more have a 0.08 limit.

ggplot(data, aes(x=as.factor(bac_combined), y=totfatrte)) + 
  geom_boxplot() +
  ggtitle('BAC level vs Fatalities') +
  xlab('BAC level') + ylab('Fatalities per 100k population')

Fatalities appear to be lowest in states with a 0.08 BAC limit and highest in states with no BAC limit.

ggplot(data, aes(x = year, y = sl_combined, group = state)) + 
  geom_jitter(alpha = 0.3) + theme_bw() +
  ggtitle("Speed limit")

Over time, speed limits have increased.

df$slhigh <- (df$sl70plus == 1)
ggplot(df, aes(x=as.factor(slhigh), y=totfatrte)) + 
  geom_boxplot() +
  ggtitle('Total Fatalities between states that have high vs low speed limit') +
  scale_x_discrete(labels=c('Low', 'High')) +
  xlab('Speed limit level') + ylab('Fatalities per 100k population')

# vehicmiles               vehicle miles traveled, billions
ggplot(data, aes(x = year, y = vehicmiles, colour = as.factor(state))) +
  geom_line(alpha = 0.7, show.legend = F) +
  ggtitle("Vehicle miles traveled, billions") + theme_bw()

Miles traveled have increased over time.

ggplot(data, aes(x=vehicmilespc, y=totfatrte)) + 
  geom_point() +
  geom_smooth(method=lm) +
  ggtitle('Vehicle miles travelled vs Fatalities') +
  xlab('Vehicle miles travelled') + ylab('Fatalities per 100k population') 

More miles travelled correlates with more fatalities.

# unem                     unemployment rate, percent
ggplot(data, aes(x = year, y = unem, colour = as.factor(state))) +
  geom_line(alpha = 0.7, show.legend = F) +
  ggtitle("Unemployment rate, percent") + theme_bw()

Increase in unemployment rate in early 80s followed by decrease until early 90s; steady decrease from mid 90s.

ggplot(data, aes(x=unem, y=totfatrte)) + 
  geom_point() +
  geom_smooth(method=lm) +
  ggtitle('Unemployement vs Fatalities') +
  xlab('Unemployment') + ylab('Fatalities per 100k population') 

Higher unemployment correlates with more fatalities.

# perc14_24          percent population aged 14 through 24
ggplot(data, aes(x = year, y = perc14_24, colour = as.factor(state))) +
  geom_line(alpha = 0.7, show.legend = F) +
  ggtitle("Percent population aged 14 through 24") + theme_bw()

Rapid decrease in percent population aged 14 through 24 until early 90s; slight increase afterwards.

ggplot(data, aes(x=perc14_24, y=totfatrte)) + 
  geom_point() +
  geom_smooth(method=lm) +
  ggtitle('Percent of Pop 14-24 vs Fatalities') +
  xlab('Percent of Pop 14-24') + ylab('Fatalities per 100k population') 

Higher percent of population 14-24 correlates with more fatalities.

data$perseround = round(data$perse)
ggplot(data, aes(x = year, y = perseround, group = state)) + 
  geom_jitter(alpha = 0.3) + theme_bw() +
  ggtitle("Per Se Laws")

States have increasingly implemented per se laws over time.

data$perseround = round(data$perse)
ggplot(data, aes(x=as.factor(perseround), y=totfatrte)) + 
  geom_boxplot() +
  ggtitle('Per Se Law vs Fatalities') +
  xlab('Per Se Law') + ylab('Fatalities per 100k population')

States with no per se laws have higher fatalities

data$gdlround = round(data$gdl)
ggplot(data, aes(x = year, y = gdlround, group = state)) + 
  geom_jitter(alpha = 0.3) + theme_bw() +
  ggtitle("Graduated Drivers License Laws")

States have increasingly implemented graduated drivers license laws beginning in the late 90s.

data$gdlround = round(data$gdl)
ggplot(data, aes(x=as.factor(gdlround), y=totfatrte)) + 
  geom_boxplot() +
  ggtitle('Graduated Drivers License Laws vs Fatalities') +
  xlab('Graduated Drivers License Law') + ylab('Fatalities per 100k population')

States with without graduated drivers license laws have higher fatalities.

data$sbprimround = round(data$sbprim)
ggplot(data, aes(x = year, y = sbprimround, group = state)) + 
  geom_jitter(alpha = 0.3) + theme_bw() +
  ggtitle("Primary Seat Belt Laws")

States have increasingly implemented primary seatbelt laws over time.

data$sbprimround = round(data$sbprim)
ggplot(data, aes(x=as.factor(sbprimround), y=totfatrte)) + 
  geom_boxplot() +
  ggtitle('Primary Seatbelt Law vs Fatalities') +
  xlab('Primary Seatbelt Law') + ylab('Fatalities per 100k population')

States with no primary seatbelt law have higher fatalities.

data$sbseconround = round(data$sbsecon)
ggplot(data, aes(x = year, y = sbprimround, group = state)) + 
  geom_jitter(alpha = 0.3) + theme_bw() +
  ggtitle("Secondary Seat Belt Laws")

States have increasingly implemented secondary seatbelt laws over time.

data$sbseconround = round(data$sbsecon)
ggplot(data, aes(x=as.factor(sbseconround), y=totfatrte)) + 
  geom_boxplot() +
  ggtitle('Secondary Seatbelt Law vs Fatalities') +
  xlab('Secondary Seatbelt Law') + ylab('Fatalities per 100k population')

States with no seconday seatbelt law have higher fatlities.

2. How is the our dependent variable of interest totfatrte defined? What is the average of this variable in each of the years in the time period covered in this dataset? Estimate a linear regression model of totfatrte on a set of dummy variables for the years 1981 through 2004. What does this model explain? Describe what you find in this model. Did driving become safer over this period? Please provide a detailed explanation.

totfatrte is defined as “nighttime fatalities per 100,000 population”

ddply(data, .(year), summarize,  Total=mean(totfatrte))

   year    Total
1  1980 25.49458
2  1981 23.67021
3  1982 20.94250
4  1983 20.15292
5  1984 20.26750
6  1985 19.85146
7  1986 20.80042
8  1987 20.77479
9  1988 20.89167
10 1989 19.77229
11 1990 19.50521
12 1991 18.09479
13 1992 17.15792
14 1993 17.12771
15 1994 17.15521
16 1995 17.66854
17 1996 17.36938
18 1997 17.61062
19 1998 17.26542
20 1999 17.25042
21 2000 16.82562
22 2001 16.79271
23 2002 17.02958
24 2003 16.76354
25 2004 16.72896

mod1 <- lm(totfatrte ~ factor(year) , data=data)
summary(mod1)

Call:
lm(formula = totfatrte ~ factor(year), data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-12.9302  -4.3468  -0.7305   3.7488  29.6498 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)       25.4946     0.8671  29.401  < 2e-16 ***
factor(year)1981  -1.8244     1.2263  -1.488 0.137094    
factor(year)1982  -4.5521     1.2263  -3.712 0.000215 ***
factor(year)1983  -5.3417     1.2263  -4.356 1.44e-05 ***
factor(year)1984  -5.2271     1.2263  -4.263 2.18e-05 ***
factor(year)1985  -5.6431     1.2263  -4.602 4.64e-06 ***
factor(year)1986  -4.6942     1.2263  -3.828 0.000136 ***
factor(year)1987  -4.7198     1.2263  -3.849 0.000125 ***
factor(year)1988  -4.6029     1.2263  -3.754 0.000183 ***
factor(year)1989  -5.7223     1.2263  -4.666 3.42e-06 ***
factor(year)1990  -5.9894     1.2263  -4.884 1.18e-06 ***
factor(year)1991  -7.3998     1.2263  -6.034 2.14e-09 ***
factor(year)1992  -8.3367     1.2263  -6.798 1.68e-11 ***
factor(year)1993  -8.3669     1.2263  -6.823 1.43e-11 ***
factor(year)1994  -8.3394     1.2263  -6.800 1.66e-11 ***
factor(year)1995  -7.8260     1.2263  -6.382 2.51e-10 ***
factor(year)1996  -8.1252     1.2263  -6.626 5.25e-11 ***
factor(year)1997  -7.8840     1.2263  -6.429 1.86e-10 ***
factor(year)1998  -8.2292     1.2263  -6.711 3.01e-11 ***
factor(year)1999  -8.2442     1.2263  -6.723 2.77e-11 ***
factor(year)2000  -8.6690     1.2263  -7.069 2.67e-12 ***
factor(year)2001  -8.7019     1.2263  -7.096 2.21e-12 ***
factor(year)2002  -8.4650     1.2263  -6.903 8.32e-12 ***
factor(year)2003  -8.7310     1.2263  -7.120 1.88e-12 ***
factor(year)2004  -8.7656     1.2263  -7.148 1.54e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.008 on 1175 degrees of freedom
Multiple R-squared:  0.1276,    Adjusted R-squared:  0.1098 
F-statistic: 7.164 on 24 and 1175 DF,  p-value: < 2.2e-16

This model suggests that driving got safer over the time period as each year has an increasing beta and all years are significant after 1981.

The summary shows that the coefficients are estimated to be negative for each year after 1980, with the coefficients being statistically significant for all years except 1981. This corroborates the observations we made in the box plot of totfatrte over time. Does this mean that driving became safer over this time period? The answer to that question is multi-faceted - better driving habits, more/less speeding, frequency of drunk driving, car safety.

3. Expand your model in Exercise 2 by adding variables bac08, bac10, perse, sbprim, sbsecon, sl70plus, gdl, perc14_24, unem, vehicmilespc, and perhaps transformations of some or all of these variables. Please explain carefully your rationale, which should be based on your EDA, behind any transformation you made. If no transformation is made, explain why transformation is not needed. How are the variables bac8 and bac10 defined? Interpret the coefficients on bac8 and bac10. Do per se laws have a negative effect on the fatality rate? What about having a primary seat belt law? (Note that if a law was enacted sometime within a year the fraction of the year is recorded in place of the zero-one indicator.)

mod2 <- lm(totfatrte ~ factor(year) + bac08 + bac10 + perse + sbprim + sbsecon + sl70plus + gdl + perc14_24 + unem + vehicmilespc, data=data)
summary(mod2)

Call:
lm(formula = totfatrte ~ factor(year) + bac08 + bac10 + perse + 
    sbprim + sbsecon + sl70plus + gdl + perc14_24 + unem + vehicmilespc, 
    data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-14.9160  -2.7384  -0.2778   2.2859  21.4203 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)      -2.716e+00  2.476e+00  -1.097 0.272847    
factor(year)1981 -2.175e+00  8.276e-01  -2.629 0.008686 ** 
factor(year)1982 -6.596e+00  8.534e-01  -7.729 2.33e-14 ***
factor(year)1983 -7.397e+00  8.690e-01  -8.512  < 2e-16 ***
factor(year)1984 -5.850e+00  8.763e-01  -6.676 3.79e-11 ***
factor(year)1985 -6.483e+00  8.948e-01  -7.245 7.82e-13 ***
factor(year)1986 -5.853e+00  9.307e-01  -6.289 4.52e-10 ***
factor(year)1987 -6.367e+00  9.670e-01  -6.585 6.87e-11 ***
factor(year)1988 -6.592e+00  1.014e+00  -6.502 1.17e-10 ***
factor(year)1989 -8.071e+00  1.053e+00  -7.667 3.68e-14 ***
factor(year)1990 -8.959e+00  1.077e+00  -8.319 2.46e-16 ***
factor(year)1991 -1.107e+01  1.101e+00 -10.052  < 2e-16 ***
factor(year)1992 -1.288e+01  1.123e+00 -11.473  < 2e-16 ***
factor(year)1993 -1.273e+01  1.136e+00 -11.204  < 2e-16 ***
factor(year)1994 -1.236e+01  1.157e+00 -10.685  < 2e-16 ***
factor(year)1995 -1.195e+01  1.184e+00 -10.098  < 2e-16 ***
factor(year)1996 -1.388e+01  1.223e+00 -11.343  < 2e-16 ***
factor(year)1997 -1.426e+01  1.250e+00 -11.408  < 2e-16 ***
factor(year)1998 -1.504e+01  1.265e+00 -11.886  < 2e-16 ***
factor(year)1999 -1.509e+01  1.284e+00 -11.750  < 2e-16 ***
factor(year)2000 -1.544e+01  1.305e+00 -11.831  < 2e-16 ***
factor(year)2001 -1.618e+01  1.334e+00 -12.131  < 2e-16 ***
factor(year)2002 -1.672e+01  1.348e+00 -12.406  < 2e-16 ***
factor(year)2003 -1.702e+01  1.359e+00 -12.521  < 2e-16 ***
factor(year)2004 -1.671e+01  1.387e+00 -12.049  < 2e-16 ***
bac08            -2.498e+00  5.375e-01  -4.648 3.73e-06 ***
bac10            -1.418e+00  3.963e-01  -3.577 0.000362 ***
perse            -6.201e-01  2.982e-01  -2.079 0.037791 *  
sbprim           -7.533e-02  4.908e-01  -0.153 0.878032    
sbsecon           6.728e-02  4.293e-01   0.157 0.875492    
sl70plus          3.348e+00  4.452e-01   7.521 1.09e-13 ***
gdl              -4.269e-01  5.269e-01  -0.810 0.417978    
perc14_24         1.416e-01  1.227e-01   1.154 0.248675    
unem              7.571e-01  7.791e-02   9.718  < 2e-16 ***
vehicmilespc      2.925e-03  9.497e-05  30.804  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.046 on 1165 degrees of freedom
Multiple R-squared:  0.6078,    Adjusted R-squared:  0.5963 
F-statistic:  53.1 on 34 and 1165 DF,  p-value: < 2.2e-16

bac8 is blood alcohol limit .08 bac10 is blood alcohol limit .10

4. Reestimate the model from Exercise 3 using a fixed effects (at the state level) model. How do the coefficients on bac08, bac10, perse, and sbprim compare with the pooled OLS estimates? Which set of estimates do you think is more reliable? What assumptions are needed in each of these models? Are these assumptions reasonable in the current context?

5. Would you perfer to use a random effects model instead of the fixed effects model you built in Exercise 4? Please explain.

6. Suppose that vehicmilespc, the number of miles driven per capita, increases by \(1,000\). Using the FE estimates, what is the estimated effect on totfatrte? Please interpret the estimate.

7. If there is serial correlation or heteroskedasticity in the idiosyncratic errors of the model, what would be the consequences on the estimators and their standard errors?